Transient Analysis of Semi—Markov Reliability Models — A Tutorial Review with Emphasis on Discrete—Parameter Approaches

Semi—Markov models can be usefully employed for the analysis of various reliability and performance characteristics of technical systems. We consider semi—Markov dependability models of systems whose finite state space S is partitioned into the set of up states u, the set of repairable down states D, and a state ω, standing for irrecoverable system failure: S = u ∪ D ∪ {ω}. A great number of reliability and performance measures for such models are examined in this chapter. It is discussed how these measures can be obtained as solutions of certain systems of integral equations if the modelling process has a continuous time parameter. The discrete parameter framework is discussed separately in more detail, since, even though it is subsumed within the continuous parameter case, it deserves special attention for three reasons. First, in some applications, discrete—time modelling is appropriate. Second, discrete—time models can be used for approximately analysing continuous—time models. Finally, for discrete—time models, the computational solution is technically less involved and it will thus be more accessible for even the mathematically less sophisticated analyst. We also present a framework within which many of the measures will be seen to admit of a formal closed form solution.